Here is a thought, I think I have found a reason why I often fail to get nice proper basses from the A4: wrong gain staging.
When I program bass patches on the A4, these have to compete, loudness wise, with what is already there in the kit. I usually start with a kick track. It is really easy to create very loud sounds with filter resonance and overdrive, so my bass patches have to compete with that, i.e. with a kick patch that is already at the maximum loudness the A4 can deliver over Overbridge, somewhere at -18db. Hence, I end up, mostly, with oscillator levels at >100, plus filter overdrive and so forth - resulting in patches that are distorted and often sound very similar to the ones I already have.
As a solution, I have started to put a +12db Utility in Ableton behind my A4 overbridge vst - the A4 is almost noise-free, so no problem with that - and bingo, I have many more options when I program patches. I now always start at OSC level 50 with all my patches, and use a limiter to sefeguard my speakers as the A4 can really throw a punch with some presets when resonance is used. For final volume of a patch, I lower or increase the volume of each track using the amp level to get to -18db peak (which is actually -18 - 12 = -30db).
Still, I think the square wave of the A4 needs some love, but I think the approach above helps to create better patches.